According to Anacreon, white roses appeared in the foam of the sea around the shell of Aphrodite's birth. The birth account of Aphrodite is first recorded in Hesod's Theogony. We all know this account from classic paintings, but what is perhaps less known is that Aphrodite was not the only one born from the casting of Ouranos' genitals in the sea by his son, and Aphrodite is not her only name:
Then the son from his ambush stretched forth his left hand and in his right took the great long sickle with jagged teeth, and swiftly lopped off his own father's members and cast them away to fall behind him. And not vainly did they fall from his hand; for all the bloody drops that gushed forth Earth received, and as the seasons moved round she bare the strong Erinyes and the great Giants with gleaming armour, holding long spears in their hands and the Nymphs whom they call Meliae all over the boundless earth. And so soon as he had cut off the members with flint and cast them from the land into the surging sea, they were swept away over the main a long time: and a white foam spread around them from the immortal flesh, and in it there grew a maiden. First she drew near holy Cythera, and from there, afterwards, she came to sea-girt Cyprus, and came forth an awful and lovely goddess, and grass grew up about her beneath her shapely feet. Her gods and men call Aphrodite, and the foam-born goddess and rich-crowned Cytherea, because she grew amid the foam, and Cytherea because she reached Cythera, and Cyprogenes because she was born in billowy Cyprus, and Philommedes because sprang from the members. And with her went Eros, and comely Desire followed her at her birth at the first and as she went into the assembly of the gods. This honour she has from the beginning, and this is the portion allotted to her amongst men and undying gods, -- the whisperings of maidens and smiles and deceits with sweet delight and love and graciousness. - Hesod -Theogones
Some of the Nymphs in these verses are also associated with flowers, and roses in particular. Aphrodite's symbols have always included the sea, conch shells, and roses. The rose and myrtle flowers were both sacred to Aphrodite.
As to the "blood and roses" story you mention, this myth of Aphrodite and Adonis is probably derived from the ancient Sumerian legend of Inanna and Dumuzid. The Greek name Ἄδωνις (Adōnis) is derived from the Canaanite word ʼadōn, meaning "lord". There are two principle versions of this story, and Ovid's work does not mention the version with the roses and the blood. I could only find a link to a study that should contain this text, but not the original:
Cyrino, Monica S. (2010), Aphrodite, Gods and Heroes of the Ancient World, New York City, New York and London, England: Routledge, ISBN 978-0-415-77523-6
In the Middle Ages, this story was transformed into a tale of platonic love, as can be found in Le Roman de la Rose (The novel of the Rose)